EDB — 17H

[17H] Topics:separation. Prerequisites:[17D]↺↻.

Given $A\subset {\mathbb{R}}^n$ closed non-empty convex and $z\notin A$, let $x^{\ast }$ be defined as in the previous exercise [17D]↺↻; define $𝛿=\Vert z-x^{\ast }\Vert$, $v=(z-x^{\ast })/𝛿$ and $a=⟨v,x^{\ast }⟩$. Prove that $v,a$ and $v,a+𝛿$ define two parallel hyperplanes that strongly separate $z$ from $A$, in the sense that $⟨z,v⟩=a+𝛿$ but $\forall x\in A,⟨x,v⟩\le a$.